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equations
Loïc Bourdin, Jacky Cresson, Isabelle Greff, Pierre Inizan
To cite this version:
EULER-LAGRANGE EQUATIONS. by
Lo¨ıc Bourdin, Jacky Cresson, Isabelle Greff & Pierre Inizan
.
Abstract. — We extend the notion of variational integrator for classical Euler-Lagrange equa-tions to the fractional ones. As in the classical case, we prove that the variational integrator allows to preserve Noether-type results at the discrete level.
Keywords: Euler-Lagrange equations; fractional calculus; variational integrator; Noether’s theorem.
AMS Classification: 70H03; 37K05; 26A33.
1. Introduction
Fractional calculus is the emerging mathematical field dealing with the generalization of the derivative to any real order. During the last two decades, it has been successfully applied to problems in economics [9], computational biology [20] and several fields in Physics [5, 6, 16, 26]. We refer to [17, 23, 25] for a general theory and to [19] for more details concerning the recent history of fractional calculus. Particularly, a subtopic of the fractional calculus has recently gained importance: it concerns the variational principles on functionals involving fractional derivatives. This leads to the statement of fractional Euler-Lagrange equations, see [1, 7, 24].
Fractional Euler-Lagrange equations are difficult to solve explicitly and consequently, it is of interest to develop efficient numerical schemes for such dynamical systems. There exists a suitable method for classical Euler-Lagrange equations called variational integrator and well-developed in [15, 21]. The basic idea is to preserve at the discrete level the intrinsic variational structure of the differential equation. In this paper, we extend the notion of variational integrator to the fractional case.
The paper is organized as follows. Section 2 is devoted to a reminder concerning fractional calculus and fractional Euler-Lagrange equations. In Section 3, we extend the notion of vari-ational integrator to the fractional case and provide numerical simulations for the fractional Dirichlet example. In Section 4, we remind the fractional Noether-type result proved in [10, 14] and we prove that it is preserved at the discrete level by the variational integrator constructed in the previous section. We conclude this section with the fractional harmonic oscillator example.
2. Reminder about fractional Euler-Lagrange equations
In this paper, we consider fractional differential systems in Rdwhere d ∈ N∗
is the dimension.
The trajectories of these systems are curves q ∈ C0([a, b], Rd) with a < b two reals. Let
0 < α ≤ 1 denote a fractional order and let us define α0 := 1 and for any r ∈ N∗:
αr :=
(−α)(1 − α) . . . (r − 1 − α)
r! . (1)
2.1. Fractional derivatives of Gr¨unwald-Letnikov. — Fractional calculus deals with
the generalization of the usual notion of derivative to any real order. Since 1695, numerous notions of fractional derivatives emerge over the year, see [17, 23, 25]. We will use the
fractional operators of Gr¨unwald-Letnikov (1867) whose definitions are reminded below.
Let q be a function defined on [a, b] with values in Rd. The left (resp. right) fractional
derivative in the sense of Gr¨unwald-Letnikov with inferior limit a (resp. superior limit b) of
order α of q is given by:
∀t ∈]a, b], Dα−q(t) := lim h→0+ nh=t−a 1 hα n X r=0 αrq(t − rh), (2) respectively: ∀t ∈ [a, b[, Dα+q(t) := lim h→0+ nh=b−t 1 hα n X r=0 αrq(t + rh), (3)
provided the right side terms are defined.
In the classical case α = 1, we note that αr= 0 for any r ≥ 2. Consequently, we obtain that
D1
− (resp. −D+1) coincides with the usual notion of left (resp. right) derivative of a function.
We remind the following result used further in the paper:
Lemma 1 (Fractional integration by parts). — For any q, w ∈ C1([a, b], Rd) satisfying
w(a) = w(b) = 0, the following equality holds:
Z b a q · D−αw dt = Z b a Dα+q · w dt. (FIBP)
Let us recall that the fractional derivatives of Gr¨unwald-Letnikov coincide with the
2.2. Fractional Euler-Lagrange equations. — Since 1997 (see [24]), a subtopic of fractional calculus gains importance: it concerns the calculus of variations on functionals
involving fractional derivatives. It leads to the statement of fractional Euler-Lagrange
equations that was developed in [1].
Let L be a Lagrangian, i.e. a C2 map of the form:
L : Rd× Rd× [a, b] −→ R
(x, v, t) 7−→ L(x, v, t)
(4) and let us consider a fractional Lagrangian functional given by:
Lα: C2([a, b], Rd) −→ R q 7−→ Z b a L(q, D−αq, t) dt. (5) Let C2
0([a, b], Rd) := {w ∈ C2([a, b], Rd), w(a) = w(b) = 0} denote the set of variations of
Lα. The critical points of Lα are the elements q ∈ C2([a, b], Rd) satisfying DLα(q)(w) = 0
for any variations w where:
DLα(q)(w) := lim
ε→0
Lα(q + εw) − Lα(q)
ε . (6)
Finally, combining Lemma 1 and an usual calculus of variations, critical points of Lα are
characterized by solutions of the following fractional Euler-Lagrange equation: ∂L ∂x(q, D α −q, t) + Dα+ ∂L ∂v(q, D α −q, t) = 0. (ELα)
We refer to [1] for a detailed proof.
3. Discrete fractional Euler-Lagrange equations
In this section, we are interested in the construction of a variational integrator for fractional
Euler-Lagrange equations (ELα). Then, let us introduce some notations available in the
whole paper. Let N ≥ 2, let h := (b − a)/N be the step size of the discretization and let
T := (tk)k=0,...,N := (a + kh)k=0,...,N be the usual partition of the interval [a, b].
3.1. Discrete fractional derivatives of Gr¨unwald-Letnikov. — In order to give a
discrete version of a fractional differential equation, we need discrete fractional operators approximating the continuous ones. As it is done in [13], we define the following left (resp.
right) discrete fractional derivative of Gr¨unwald-Letnikov:
∆α− (resp. ∆α+) is an approximation of Dα−(resp. D+α). We refer to [12, 15] for details about
these approximations of fractional operators.
In the classical case α = 1, we note that ∆1
−and −∆1+coincide respectively with the backward
and forward Euler’s approximations of the derivative d/dt. Indeed, we have:
∆1 −: (Rd)N +1 −→ (Rd)N Q= (Qk)k=0,...,N 7−→ Qk − Qk−1 h k=1,...,N , (9) and: ∆1 +: (Rd)N +1 −→ (Rd)N Q= (Qk)k=0,...,N 7−→ Qk− Qk+1 h k=0,...,N −1 . (10)
3.2. Remark about direct discretization and variational integrator. — An usual and algebraic way to obtain a discrete version of a fractional differential equation is to replace
the curves q ∈ C0([a, b], Rd) by discrete elements Q ∈ (Rd)N +1 and to replace fractional
derivatives Dα± by discrete fractional derivatives ∆α±. This method is widely used and one
can find examples in [4, 18, 22] for fractional differential equations and fractional partial differential equations. Other examples can be found in [2, 3, 11] for fractional optimal control problems.
Such a discretization leads to the following numerical scheme for a fractional Euler-Lagrange
equation (ELα): ∀k = 1, . . . , N − 1, ∂L ∂x Qk, (∆ α −Q)k, tk + ∆α+ ∂L ∂v(Q, ∆ α −Q, T ) k = 0. (11)
However, a fractional Euler-Lagrange equation (ELα) admits a variational structure in the
sense that it derives from a variational principle on a functional, see Section 2.2. This structure is intrinsic and induces strong constraints on the qualitative behaviour of the solutions. It seems then important to preserve this structure at the discrete level. Nev-ertheless, the numerical scheme (11) is obtained by an algebraic procedure only based on
the differential writing of (ELα). Consequently, there are no guarantees that the intrinsic
variational structure of (ELα) is preserved at the discrete level.
In Section 3.3, we are going to construct a variational integrator for fractional Euler-Lagrange
equations (ELα). This procedure has a variational approach and allows to preserve the
vari-ational structure of (ELα) at the discrete level.
Let us give more details concerning the construction of a variational integrator. Let us consider a differential system admitting a variational structure (i.e. deriving from a variational principle on a functional and then, characterizing its critical points). A variational integrator is a numerical scheme constructed as follows:
– the first step consists in defining a discrete version of the functional;
Hence, a numerical scheme is obtained and it is called variational integrator. The variational structure of the differential system is then preserved at the discrete level in the sense that the discrete solutions correspond to the discrete critical points of the discrete version of the initial functional.
We recall that a variational integrator is well-studied for classical Euler-Lagrange equations (α = 1) in [15, 21]. In this case, the conservation of the variational structure at the discrete level allows to preserve properties and results relative to this structure. As an example, we can cite Noether’s theorem. In the following section, our aim is to extend this variational integrator to the fractional case (0 < α ≤ 1).
3.3. Construction of a variational integrator. — In this section, we construct a varia-tional integrator for fracvaria-tional Euler-Lagrange equations following the two steps described in the previous section.
First step. — Let us define a discrete version of the fractional Lagrangian functional Lα.
Considering the usual Gaussian quadrature formula, we define the following discrete fractional Lagrangian functional: Lαh : (Rd)N +1 −→ R Q 7−→ h N X k=1 L Qk, (∆α−Q)k, tk. (12)
Hence, the first step of construction of a variational integrator for (ELα) is completed.
Second step. — Let (Rd)N +10 := {W ∈ (Rd)N +1, W0= WN = 0} denote the set of discrete
variations of Lα
h. In the sequel, we focus on discrete critical points of Lαh, i.e. elements
Q∈ (Rd)N +1satisfying DLαh(Q)(W ) = 0 for any discrete variations W where:
DLαh(Q)(W ) := lim
ε→0
Lαh(Q + εW ) − Lαh(Q)
ε . (13)
As in the continuous case, in order to obtain a characterization of the discrete critical points
of Lαh, we first need a preliminary result. We prove the following discrete version of Lemma 1:
Lemma 2 (Discrete fractional integration by parts). — For any Q, W ∈ (Rd)N +1
satisfying W0= WN = 0, the following equality holds:
Proof. — Since W0 = WN = 0, the following equalities hold: hα N X k=1 Qk· (∆α−W)k = N X k=1 k X r=0 αrQk· Wk−r = N X k=0 k X r=0 αrQk· Wk−r = N X r=0 N X k=r αrQk· Wk−r = N X r=0 N −r X k=0 αrQk+r· Wk = N X k=0 N −k X r=0 αrQk+r· Wk = N −1 X k=0 N −k X r=0 αrQk+r· Wk = hα N −1 X k=0 (∆α+Q)k· Wk. (14)
Multiplying by h1−α, the proof is complete.
Finally, combining Lemma 2 and a discrete calculus of variations, we obtain the following result:
Theorem 1. — Let Q ∈ (Rd)N +1. Then, Q is a discrete critical point of Lα
h if and only if
Qis solution of the following discrete fractional Euler-Lagrange equation:
∀k = 1, . . . , N − 1, ∂L ∂x Qk, (∆ α −Q)k, tk + ∆α+ ∂L ∂v(Q, ∆ α −Q, T ) k = 0. (ELαh)
Proof. — Let Q ∈ (Rd)N +1 and W ∈ (Rd)N +1
0 . Let us define the following function:
ϕ : R −→ R ε 7−→ Lαh(Q + εW ) = h N X k=1 L Qk+ εWk, (∆α−Q)k+ ε(∆α−W)k, tk. (15)
Then, since DLαh(Q)(W ) = ˙ϕ(0), we have:
DLαh(Q)(W ) = h N X k=1 ∂L ∂x Qk, (∆ α −Q)k, tk · Wk+ ∂L ∂v Qk, (∆ α −Q)k, tk · (∆α−W)k . (16)
Since W ∈ (Rd)N +10 and using the discrete fractional integration by parts given in Lemma 2,
we obtain: DLαh(Q)(W ) = h N −1 X k=1 ∂L ∂x Qk, (∆ α −Q)k, tk + ∆α+ ∂L ∂v(Q, ∆ α −Q, T ) k · Wk, (17)
which completes the proof.
Theorem 1 completes the second step. Precisely, the discrete fractional Euler-Lagrange
equa-tion (ELαh) is the variational integrator constructed for the fractional Euler-Lagrange equation
(ELα). It is a numerical scheme preserving the variational structure of (ELα) in the sense
that the discrete solutions of (ELαh) coincide with the discrete critical points of the discrete
version Lα
h of Lα.
Remark 1. — We note that the discrete Euler-Lagrange equation (ELα
h) coincides with the
numerical scheme (11) obtained with a direct discretization. It is important to note that such
a phenomena is not obvious. Indeed, in the classical case α = 1, we have Dα
and then, it is not obvious to know how to replace d/dt at the discrete level. Indeed, one can
choose ∆1
− or −∆1+ or a mixing of the two of them. In the fractional case, the non locality of
the fractional operators does not permit such a choice.
In the classical case α = 1, we note that (ELαh) coincides with the classical discrete
Euler-Lagrange equation provided in literature about variational integrators, see [15, 21].
3.4. The fractional Dirichlet example. — In general, the fractional Euler-Lagrange equations are very difficult to solve explicitly. There are only few examples where exact solu-tions are provided and furthermore not in an explicit way. Due to the additional complexity of fractional operators, the purpose of this section is only to obtain experimental results and not to provide numerical analyses.
In this section, we consider the fractional Dirichlet example studied in [1] and where a quasi-explicit solution is known. Precisely, we consider the interval [a, b] = [0, 1], d = 1 and the following Lagrangian: L : R × R × [0, 1] −→ R (x, v, t) 7−→ 1 2v 2. (18)
The associated fractional Euler-Lagrange equation (ELα) is given by Dα
+◦ D−αq = 0. For any
1/2 < α ≤ 1, under boundary assumptions q(0) = 0 and q(1) = 1, this equation admits an unique solution given by:
q(t) = (2α − 1)
Z t
0
1
[(1 − x)(t − x)]1−α dx. (19)
We note that for α = 1, the unique solution is q(t) = t. Nevertheless, in the strict fractional case 1/2 < α < 1, q is not explicitly given since it is written with an improper integral. As a consequence, the following numerical results consider an approximation of the exact solution q using high-order global adaptive quadrature.
Let us solve the discrete fractional Euler-Lagrange equation (ELαh) given by:
∀k = 1, . . . , N − 1, (∆α+◦ ∆α−Q)k = 0, (20)
with boundary values Q0 = 0 and QN = 1. For any 1/2 < α ≤ 1 and any N ≥ 2, the ℓ∞-error
and the ℓ2-error between the exact solution q and the discrete one Q are defined by:
where h = 1/N . We compare q and Q for α = 3/4 and N = 100. The solutions are displayed on the following picture:
0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 alpha=3/4 and N=100 Q q The errors E∞
α,N and Eα,N2 for α = 3/4 and for varying N are given in the following table:
N 50 100 200 250 500 1000 2000 4000
E∞
3/4,N 0.0255 0.0185 0.0134 0.0120 0.0086 0.0062 0.0044 0.0031
E2
3/4,N 0.0140 0.0100 0.0072 0.0064 0.0046 0.0032 0.0023 0.0016
Finally, the graphic representations of log(E∞
3/4,N) and log(E3/4,N2 ) with respect to log(h) are
respectively given on the following pictures:
−8.5 −8 −7.5 −7 −6.5 −6 −5.5 −5 −4.5 −4 −3.5 −6 −5.5 −5 −4.5 −4 −3.5 log(h) log(E ∞ α),N −8.5 −8 −7.5 −7 −6.5 −6 −5.5 −5 −4.5 −4 −3.5 −6.5 −6 −5.5 −5 −4.5 −4 log(h) log(E 2)α,N
We clearly obtain two linear functions with the common slope λ3/4 ≃ 0.51. In this case, we
conclude that we obtain an experimental convergence of order λ3/4≃ 0.51.
Now, let us study the evolution of the errors E∞
α,N and E2α,N and the slope λαwith respect to
α. Let us precise that the numerical study of the case α = 1 is not interesting because, in this
case, the discrete Euler-Lagrange equation (EL1
h) gives directly the exact solution q(t) = t.
In the strict fractional case 1/2 < α < 1, the errors E∞
α,N and Eα,N2 are given in the following
α\N 50 100 200 250 500 1000 2000 4000 0.55 0.3811 0.3679 0.3527 0.3475 0.3310 0.3141 0.2974 0.2810 0.1746 0.1560 0.1393 0.1344 0.1203 0.1079 0.0972 0.0878 0.6 0.1988 0.1788 0.1596 0.1537 0.1363 0.1204 0.1061 0.0933 0.0969 0.0825 0.0701 0.0665 0.0566 0.0483 0.0413 0.0354 0.65 0.1023 0.0856 0.0711 0.0669 0.0552 0.0453 0.0372 0.0304 0.0525 0.0423 0.0340 0.0317 0.0255 0.0205 0.0165 0.0133 0.7 0.0517 0.0403 0.0311 0.0286 0.0219 0.0168 0.0129 0.0098 0.0276 0.0210 0.0159 0.0146 0.0110 0.0083 0.0063 0.0048 0.75 0.0255 0.0185 0.0134 0.0120 0.0086 0.0062 0.0044 0.0031 0.0140 0.0100 0.0072 0.0064 0.0046 0.0032 0.0023 0.0016 0.8 0.0122 0.0083 0.0056 0.0049 0.0033 0.0022 0.0015 0.00097 0.0067 0.0045 0.0030 0.0027 0.0018 0.0012 0.0008 0.00052 0.85 0.0055 0.0035 0.0022 0.0019 0.0012 0.00074 0.00046 0.00028 0.0030 0.0019 0.0012 0.0010 0.0006 0.0004 0.00025 0.00015 0.9 0.0022 0.0013 0.00079 0.00066 0.00039 0.00022 0.00013 0.000075 0.0012 0.0007 0.00041 0.00035 0.0002 0.00012 0.00007 0.00004 0.95 0.00068 0.00038 0.00021 0.00017 0.000095 0.000052 0.000028 0.000015 0.00037 0.0002 0.00011 0.00009 0.00005 0.000026 0.000014 0.000007
We clearly see that the more α is close to 1, the more the approximation is sensibly better.
Now, we study the evolution of slope λα with respect to α:
α 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
λα 0.11 0.2 0.3 0.39 0.51 0.62 0.72 0.8 0.91
For this example, we conclude that we obtain an experimental convergence of order λα =
2α − 1.
4. Conservation of a fractional Noether-type result
It is well-known that the conservation at the discrete level of the variational structure allows to preserve some properties relative to this structure. For example, in the classical case α = 1, we know that the classical Noether’s theorem is preserved, see [15, 21]. In this section, in the fractional case 0 < α ≤ 1, we prove the conservation at the discrete level of the fractional Noether-type result proved simultaneously by Cresson [10] and Torres et al. [14].
4.1. Reminder about a fractional Noether-type theorem. — We first review the definition of a one parameter group of diffeomorphisms:
Definition 1. — For any real s, let φ(s, ·) : Rd −→ Rd be a diffeomorphism. Then,
Φ = {φ(s, ·)}s∈R is a one parameter group of diffeomorphisms if it satisfies
1. φ(0, ·) = IdRd, 2. ∀s, s′ ∈ R, φ(s, ·) ◦ φ(s′ , ·) = φ(s + s′ , ·) , 3. φ is of class C2.
The action of a one parameter group of diffeomorphisms on a Lagrangian allows to define the notion of a symmetry for a fractional Euler-Lagrange equation:
Definition 2. — Let Φ = {φ(s, ·)}s∈R be a one parameter group of diffeomorphisms and let
L be a Lagrangian. L is said to be D−α-invariant under the action of Φ if it satisfies:
∀q solution of (ELα), ∀s ∈ R, Lφ(s, q), D−α φ(s, q), t
= L q, Dα−q, t. (22)
For example, any quadratic Lagrangian is D−α-invariant under the action of rotations. We
refer to Section 4.3 for the fractional harmonic oscillator example. We also note that the fractional Dirichlet example studied in Section 3.4 is associated to a quadratic Lagrangian and then, it admits a symmetry.
With this definition of symmetry, Cresson [10] and Torres et al. [14] proved the following result:
Theorem 2 (Fractional Noether-type theorem). — Let L be a Lagrangian Dα
−
-invariant under the action of a one parameter group of diffeomorphisms Φ = {φ(s, ·)}s∈R.
Then, the following equality holds for any solution q of (ELα):
D−α ∂φ ∂s(0, q) ·∂L ∂v(q, D α −q, t) − ∂φ ∂s(0, q) · D α + ∂L ∂v(q, D α −q, t) = 0. (23)
In the classical case α = 1, the classical Leibniz formula allows to rewrite (23) as the derivative of a product. Precisely, Theorem 2 leads to the classical Noether’s theorem given by:
Theorem 3 (Classical Noether’s theorem). — Let L be a Lagrangian d/dt-invariant
under the action of a one parameter group of diffeomorphisms Φ = {φ(s, ·)}s∈R. Then, the
following equality holds for any solution q of (EL1):
d dt ∂φ ∂s(0, q) · ∂L ∂v(q, ˙q, t) = 0, (24)
where ˙q is the classical derivative of q, i.e. dq/dt.
Theorem 3 provides an explicit constant of motion for any classical Euler-Lagrange equations
(EL1) admitting a symmetry. In the fractional case, such a simple Leibniz formula allowing
to rewrite (23) as a total derivative with respect to t is not known yet. Nevertheless, from Theorem 2, we prove in [8] the existence of an explicit constant of motion written with an infinite sum (this result is proved via an iterative application of Leibniz formulas). For sake of brevity, we do not develop this result in this paper.
4.2. A discrete fractional Noether-type theorem. — In this section, we prove that Theorem 2 is preserved at the discrete level with the discrete fractional Euler-Lagrange
equations (ELαh).
Definition 3. — Let Φ = {φ(s, ·)}s∈R be a one parameter group of diffeomorphisms and let
L be a Lagrangian. L is said to be ∆α−-invariant under the action of Φ if it satisfies:
∀Q solution of (ELαh), ∀s ∈ R, Lφ(s, Q), ∆α− φ(s, Q), T
= L Q, ∆α−Q, T. (25)
As in the continuous case, any quadratic Lagrangian is ∆α−-invariant under the action of
rotations. We refer to Section 4.3 for the fractional harmonic oscillator example. The frac-tional Dirichlet example studied in Section 3.4 is also associated to a quadratic Lagrangian and then, it admits a discrete symmetry.
From this notion of discrete symmetry, we prove the discrete analogous version of Theorem 2:
Theorem 4 (Discrete fractional Noether-type theorem). — Let L be a Lagrangian
∆α
−-invariant under the action of a one parameter group of diffeomorphisms Φ = {φ(s, ·)}s∈R.
Then, the following equality holds for any solution Q of (ELαh):
∆α− ∂φ ∂s(0, Q) ·∂L ∂v(Q, ∆ α −Q, T ) − ∂φ ∂s(0, Q) · ∆ α + ∂L ∂v(Q, ∆ α −Q, T ) = 0. (26)
Proof. — This proof is a direct adaptation to the discrete case of the proof of Theorem 2. Let
Q∈ (Rd)N +1be a solution of (ELα
h). Let us differentiate equation (25) with respect to s and
invert the operators ∆α−and ∂/∂s. We finally obtain for any s ∈ R and any k ∈ {1, . . . , N −1}:
∆α− ∂φ ∂s(s, Q) k ·∂L ∂v φ(s, Qk), ∆α− φ(s, Qk), tk +∂L ∂x φ(s, Qk), ∆α− φ(s, Qk), tk ·∂φ ∂s(s, Qk) = 0. (27)
Since φ(0, ·) = IdRd and Q is solution of (ELα
h), taking s = 0 in (27) leads to (26).
From this last result, we conclude that the discrete fractional Euler-Lagrange equation (ELαh)
allows to preserve the fractional Noether’s Theorem 2 at the discrete level. We recall the following discrete Leibniz formula:
∀F , G ∈ (Rd)N +1, ∀k = 1, . . . , N − 1, ∆1− F · σ(G)
k= (∆
1
−F)k· Gk− Fk· (∆1+G)k, (28)
where σ(G)k = Gk+1 for any k = 0, . . . , N − 1. From this discrete Leibniz formula, we note
that Theorem 4 takes a particular simple expression in the classical case α = 1. It corresponds to the discrete version of Theorem 3:
Theorem 5 (Discrete classical Noether’s theorem). — Let L be a Lagrangian ∆1
−
-invariant under the action of a one parameter group of diffeomorphisms Φ = {φ(s, ·)}s∈R.
Then, the following equality holds for any solution Q of (EL1
h): ∆1− ∂φ ∂s(0, Q) · σ ∂L ∂v(Q, ∆ 1 −Q, T ) = 0. (29)
This result is a reformulation of the discrete Noether’s theorem proved in [15, 21]. We recall the following implication:
Consequently, as in the continuous case, Theorem 5 provides an explicit discrete constant of
motion for any discrete classical Euler-Lagrange equations (EL1
h) admitting a symmetry. In
the fractional case, such a simple Leibniz formula allowing to rewrite (26) as a total discrete derivative is not known yet. Nevertheless, from Theorem 4, we prove in [8] the existence of a discrete constant of motion computable in a finite number of steps. For sake of brevity, we do not develop this result in this paper.
4.3. The fractional harmonic oscillator example. — We conclude this section with the fractional harmonic oscillator example where Noether-type Theorems 2 and 4 are available. Precisely, we consider [a, b] = [0, 1], d = 2 and the following quadratic Lagrangian:
L : R2× R2× [0, 1] −→ R
(x, v, t) 7−→ 1
2(kxk
2+ kvk2).
(31)
Since L is quadratic, for any 0 < α ≤ 1, L is Dα
−-invariant and ∆α−-invariant under the action
of rotations given by:
φ : R× R2 −→ R2 (s, x1, x2) 7−→ cos(s) − sin(s) sin(s) cos(s) x1 x2 . (32)
In the classical continuous case α = 1, the Euler-Lagrange equation (EL1) is given by ¨q = q
and consequently, the exact solutions are given by q(t) = (q1(t), q2(t)) = (c
1et+ c2e−t, c3et+
c4e−t) for any t ∈ [0, 1] where c1, c2, c3, c4 ∈ R. Applying Theorem 3, we obtain an explicit
constant of motion. Indeed, for any q = (q1, q2) solution of (EL1), there exists c ∈ R such
that: ∂φ ∂s(0, q) · ∂L ∂v(q, ˙q, t) = (−q 2, q1) · ( ˙q1, ˙q2) = c. (33)
Similarly, the discrete Euler-Lagrange equation (EL1
h) is given by −∆1+ ◦ ∆1−Q = Q and
Theorem 5 leads to an explicit discrete constant of motion. Indeed, for any Q = (Q1, Q2)
discrete solution of (EL1
h), there exists c ∈ R such that:
∂φ ∂s(0, Q) · σ ∂L ∂v(Q, ∆ 1 −Q, T ) = (−Q2, Q1) · σ(∆1−Q1, ∆1−Q2) = c. (34)
In the strict fractional continuous case 0 < α < 1, the fractional Euler-Lagrange equation
(ELα) is given by q + Dα+◦ Dα−q = 0. Even if explicit solutions are not known yet, the
fractional Noether-type Theorem 2 is available and leads to an explicit constant of motion
in [8]. In the discrete case, the discrete fractional Euler-Lagrange equation (ELα
h) is given
by Q + ∆α+◦ ∆α−Q = 0. The discrete solutions are computable and the discrete fractional
Noether-type Theorem 4 (leading to an explicit discrete constant of motion in [8]) is available. References
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Lo¨ıc Bourdin, Laboratoire de Math´ematiques et de leurs Applications - Pau (LMAP). UMR CNRS 5142. Universit´e de Pau et des Pays de l’Adour. • E-mail : [email protected]
Jacky Cresson, Laboratoire de Math´ematiques et de leurs Applications - Pau (LMAP). UMR CNRS 5142. Universit´e de Pau et des Pays de l’Adour. • E-mail : [email protected]
Isabelle Greff, Laboratoire de Math´ematiques et de leurs Applications - Pau (LMAP). UMR CNRS 5142. Universit´e de Pau et des Pays de l’Adour. • E-mail : [email protected]
